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A052991 Expansion of (1-x-x^2)/(1-3x-x^2).

%I A052991 #45 Jul 02 2025 16:01:58
%S A052991 1,2,6,20,66,218,720,2378,7854,25940,85674,282962,934560,3086642,
%T A052991 10194486,33670100,111204786,367284458,1213058160,4006458938,
%U A052991 13232434974,43703763860,144343726554,476734943522,1574548557120,5200380614882,17175690401766,56727451820180
%N A052991 Expansion of (1-x-x^2)/(1-3x-x^2).
%H A052991 Harvey P. Dale, <a href="/A052991/b052991.txt">Table of n, a(n) for n = 0..1000</a>
%H A052991 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1066">Encyclopedia of Combinatorial Structures 1066</a>
%H A052991 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,1)
%F A052991 G.f.: (-1+x+x^2)/(-1+3*x+x^2).
%F A052991 Recurrence: {a(0)=1, a(1)=2, a(n)+3*a(n+1)-a(n+2), a(2)=6}.
%F A052991 Sum(-2/13*(3*_alpha-2)*_alpha^(-1-n), _alpha=RootOf(-1+3*_Z+_Z^2)).
%F A052991 a(n) = Sum_{k=0..n} A155161(n,k)*2^k. - _Philippe Deléham_, Feb 08 2012
%F A052991 G.f.: 1/Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2013
%F A052991 a(n) = A006190(n+1)-A006190(n)-A006190(n-1). - _R. J. Mathar_, Feb 27 2019
%F A052991 a(n) = 2*A006190(n) for n>=1. - _Philippe Deléham_, Mar 09 2023
%p A052991 spec := [S,{S=Sequence(Prod(Sequence(Union(Prod(Z,Z),Z)),Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t A052991 CoefficientList[Series[(1-x-x^2)/(1-3x-x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,1},{1,2,6},30] (* _Harvey P. Dale_, May 10 2022 *)
%Y A052991 Cf. A006190, A155161.
%K A052991 easy,nonn
%O A052991 0,2
%A A052991 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052991 More terms from _James Sellers_, Jun 06 2000